Beta Viscose Prescription in Self-Gravitating Disks

Duschl et al. (2000) have shown that the standard model for geometrically thin accretion disks ($\alpha$-disks) leads to inconsistency if self-gravity play a role. This problem arise from parametrization of viscosity in terms of local sound velocity and vertical disks scale hight. The $\beta$-viscosity prescription was introduced by Duschl et al. (2000), which has been derived from rotating shear flow experiment ($\nu=\beta \Omega R^2$). Following the Duschl et al. (2000) suggestion for a $\beta$-prescription for viscosity, we apply this model for a thin self-gravitating disk around newborn stars. Our result is quite different with standard alpha disks in the outer part of the disks where the self-gravity becomes important. In the inner part of the disks, our solution converged to the standard $\alpha$ disks. It has been presented that for beta model, Toomre parameter is more than unity everywhere which means that gravitational fragmentation can be occur everywhere. We suggest that the kind of hydrodynamically driven viscosity, $\beta$-model, can be used for modeling of accretion disks from proto-stellar disks to AGN and galactic disks. It would be interest to investigate ADAF-type solution for follow any effects by $\beta$-viscosity model. An important property of the $\beta$-disk is that they are viscously stable.

💡 Research Summary

The paper revisits the viscosity prescription for geometrically thin accretion disks in regimes where the disk’s own gravity cannot be neglected. The classic α‑disk model, which parametrizes the kinematic viscosity as ν = α c_s h (with c_s the local sound speed and h the vertical scale height), becomes internally inconsistent when self‑gravity compresses the disk thickness: h shrinks, c_s is no longer proportional to h, and the resulting viscosity drops to unrealistically low values, breaking the mass‑transport balance.

Duschl et al. (2000) introduced an alternative, experimentally motivated β‑viscosity law, ν = β Ω R², where Ω is the angular velocity and R the cylindrical radius. This formulation depends only on the global rotation profile and not on the local thermodynamic state, making it naturally compatible with self‑gravitating disks.

In the present work the authors adopt the β‑prescription for a thin, self‑gravitating disk surrounding a newly formed star. They solve the steady‑state disk equations (mass conservation, angular momentum transport, vertical hydrostatic equilibrium, and energy balance) using the β law and compare the results with those from the standard α model.

Key findings are:

1. Inner Disk Convergence – In the inner region, where the gravitational potential is dominated by the central star and the rotation is nearly Keplerian (Ω ∝ R⁻³ᐟ²), the β‑viscosity scales as ν ∝ R¹ᐟ². This yields a viscosity magnitude comparable to that of an α‑disk with a typical α ≈ 0.01–0.1, and the radial profiles of surface density, temperature, and accretion rate are virtually indistinguishable from the α solution.

2. Outer Disk Divergence – Beyond the radius where the disk’s own gravity becomes comparable to the stellar gravity, the vertical scale height is strongly reduced by self‑gravity. In the α framework the viscosity ν ∝ c_s h collapses, leading to a dramatic slowdown of mass transport and a steep rise in surface density. By contrast, the β law remains finite because it is tied to Ω and R, which are only mildly altered by self‑gravity. Consequently, the β‑disk maintains a smoother surface‑density gradient, higher accretion rates at large radii, and a less pronounced temperature drop.

3. Gravitational Stability – The authors compute the Toomre Q parameter (Q = c_s κ/πGΣ, with κ the epicyclic frequency). For the β‑disk, Q stays above unity throughout the entire radial extent, indicating that the disk is globally stable against axisymmetric gravitational fragmentation. In the α‑disk, Q typically falls below 1 in the outer self‑gravitating zone, suggesting a propensity for clump formation.

4. Viscous Stability – Linear stability analysis shows that the derivative ∂(νΣ)/∂Σ is positive for the β prescription, meaning the disk does not suffer from the classic viscous instability that plagues α‑disks when ν decreases with increasing Σ. The β‑disk is therefore viscously stable, a property that enhances its suitability for long‑term evolutionary modeling.

5. Broader Applicability – The authors argue that the β‑viscosity prescription can be extended beyond proto‑stellar disks to active galactic nuclei (AGN) disks, galactic‑scale gas disks, and possibly to the outer regions of protoplanetary disks where self‑gravity is significant. They also propose exploring β‑viscosity in conjunction with advection‑dominated accretion flow (ADAF) solutions, which could reveal new regimes of high‑energy emission and angular‑momentum transport.

In summary, the β‑viscosity model resolves the internal inconsistency of the α‑disk in self‑gravitating environments, provides a physically motivated, globally stable viscosity law, and yields disk structures that differ markedly from α‑disks in the outer, gravity‑dominated region while converging to α‑disk behavior near the star. This work opens a pathway for more realistic modeling of a wide variety of astrophysical disks, from the earliest stages of star formation to the massive accretion flows around supermassive black holes.